APBRmetrics

[josh]

Hello everyone, I'm brand new here and admittedly not as up to speed on basketball as I am on stats. I like basketball, but I will be the first to admit I like stats in general more. But I really like applying stats to sports and competitions in general. I mention this so you'll forgive any statements that show my ignorance in the area.

Has anyone tried using Bayesian linear regression to solve for the adjusted plus/minus values?

I see two benefits to a Bayesian treatment:

1) It handles uncertainty well. This means, the more data there is on a player, the more confidence in that player's rating. This allows players much less play time to be included since uncertainty about their plus/minus values will be handled naturally.

2) Let's say for example we use Dan's likelihood with a multivariate normal prior on the adjusted plus/minus values. Since this prior is conjugate to the normal likelihood in standard linear regression, it allows for efficient sequential updating. A new estimate for each player can be created after *every* substitution. It would require storing covariances between every player, but that's still relatively small on today's computers. The nice thing is, the updates would require at most inverting a 10x10 instead of, say, 700x700 matrix since only 10 players participate in each update. This would mean adjusted plus/minus values could be updated in real time after every substitution.

The prior and noise variance parameters can be inferred using either a hierarchical Bayesian model or (most Bayesians from the stats community will shudder) evidence maximization a.k.a. empirical Bayes, applied to an early data sample or season.

This would let sites like Basketball Value easily post changes to Bayesian adjusted plus/minus values after every game (in theory after every substitution if the data were available real time) along with home field advantages for each court. This could also include a variance estimate on each player's plus/minus value. Also, the covariance values would give insights into which players play well together, whether opponents or not.

To be fair, and as many of you may already know, I should mention that having fast sequential updates like this is not solely Bayesian. The data could also be fit sequentially using stochastic least mean squares (LMS).

Thanks in advance for your patience with my ignorance as a newcomer to these forums.

[Ed Küpfer]

This is one of those things that sounds like a good idea, but....

The standard errors on a single season of player +/- regressions are very large. It's worthwhile to look at the numbers produced -- but those coefficients are useful mostly as descriptive of past performance. I have not seen any predictve value of these numbers based on within-season production. The amount of work involved in producing undated coefiicents on the fly dwarfs the potential reward, I think.
_________________
ed

[josh]

[Statman]

[basketballvalue]

[josh]

[Mountain]

Aaron, news of your plans to produce adjusted data for 06-07 and 07-08 in real time is welcome.

Not sure if you saw the question in the defense thread but any way you can provide offense/defense splits of the adjusted player impacts?

[Mountain]

Josh, any comments on the various papers using Bayesian techniques mentioned on the forum recently? Have you done anything new applying them to players? Just curious.

[josh]

Yeah, I've been trying several models lately, and doing some sampling (MCMC).

The advisor of the student on the BYU paper, Shane Reese, is the guy I learned a large portion of my Bayesian analysis techniques from. I was a CS student at BYU and sat in on Shane's Bayesian Data Analysis class.

Did you have any particular questions? I understand what they did pretty well, and I've used a similar approach but for +/- ratings. I should be able to release my results soon enough, but I have to investigate a few more things first.

[Mountain]

I'll look forward to seeing your work.

As for questions, beyond the detailed ones out there in the various threads I am wondering if anyone has any overarching comments about how much credit the immediately prior action should get / is getting in various systems of the credit for the next action. If that makes sense or has any use in the discussion.

Two specific questions, if you are willing:

1. What did you think of this study?

http://tinyurl.com/2yomsk
Environmental factors affecting Sam Cassell's shooting behavior and results

Is a next frontier to break apart the components of a single play with Bayesian techniques and other tools?

2. Did you have any comments on Kenny Shirley's Markov model or findings?

[josh]

When you're talking about prior action are you referring to Markov-Chain models like Shirley's? Where what happened in the previous play can affect the next?

Each data point in the model I am using is a sequence without substitutions, and so can include a number of plays.

I could model additional effects within there.

Like Reese I'm using hierarchical Bayesian linear regression.

I believe Liu uses Poisson regression, but I haven't dug too deep into the paper yet.

[Mountain]

"When you're talking about prior action are you referring to Markov-Chain models like Shirley's? Where what happened in the previous play can affect the next?"

Yes, I was interested in that general topic and his work in particular and got around to saying both in my post eventually.

"Each data point in the model I am using is a sequence without substitutions, and so can include a number of plays.

I could model additional effects within there."

Interesting, will wait to hear more.


I try to pull stuff out of those weightier studies but it helps when those familiar with the techniques provide reaction. If you got anything notable out of Liu's study I'd appreciate a summary.

P.S. Did you see this thread
http://sonicscentral.com/apbrmetrics/viewtopic.php?t=1498&start=105
And Gabe's comment on Berri's response to first question?

http://www.stat.columbia.edu/~cook/movabletype/archives/2006/06/

"AG: 1. Reading Gladwell's article, I assume that Berri et al. are doing regression analysis, i.e., estimating player abilities as a linear combination of individual statistics. I have the same question that Bill James asked in the context of baseball statistics: why restrict to linear functions? A function of the form A*B/C (that's what James used in his runs created formula, or more fully, something like (A1 + A2 +...)*(B1 + B2 +...)/C) could make more sense."

Anything further on this question? From anyone?

(I havent explored this blog but it seems like it might be crack or crackerjack for an applied stat junkie.)

[josh]

I realized I let my fingers speak without consulting my brain in that "Let Reese" sentence fragment. I think I've cleaned it up.

[Mountain]

Talk about position impacts, skill sets, play sequences and interactions, player interactions, model function types, etc. jogged my memory of a few past discussions.

These threads have discussion that I looked back at and think could be germaine
http://sonicscentral.com/apbrmetrics/viewtopic.php?t=1251
http://sonicscentral.com/apbrmetrics/viewtopic.php?t=1268